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Gravitational radiation damping of nongravitational motion
ANNALS
OF
PHYSICS:
10,
94-99
(1960)
Gravitational
Radiation
Damping
Nongravitational
Motion*
ASHER PERES AND NATHAN Department
of Physics,
Israel
of
Institute
ROSEN
of Technology,
Haifa,
Israel
A material system acted upon by internal forces much larger than the gravitational ones is considered, and the rate of work of this system against its own gravitational field is computed. The result is equal to the rate of radiated energy obtained from the linearized theory.
The existence of gravitational radiation has long been a controversial subject. It seemedwell established according to the wave zone approximation of the linearized theory (I ), but the investigation of the equations of motion (Z-8) led to very dif7erent results. Only recently, an agreement between both methods was reached in the case of freely gravitating pole particles (9) and in a special case of constrained motion (IO). Our purpose in the present paper is to show that if a system is acted on by internal forces much larger than those due to gravitation, its rate of work against its own gravitational field (averaged over a cycle of the motion or over a long time interval) is equal to the rate of loss of energy by gravitational radiation, as calculated by means of the linear approximation. The proof will be rather general, i.e., independent of the nature of the motion (rotational, vibrational, or other), but with the condition that the system remain localized within a finite volume. We take as field variables the contravariant densities Q’“, and choose quasiGalilean coordinates (Q” - q’“) subject to the harmonic conditionl: PY Q ,Y - 0. (1) The Einstein field equations then have the form, in natural units, v2Q”y _ ‘;1’Y = -16~gX:“”
+ O”‘,
(2)
where 5“’ is the stress-energy tensor density, Q = (-det. Q’“)~‘~, and 0’” is quadratic in (9”’ - r]“). The compatibility of Eqs. (1) and (2) follows from * Partly supported by the U. S. Air Force, through search and Development Command. 1 Greek indices run from 0 to 3, Latin indices-from 94
the European 1 to 3.
Office of the Air Re-
GRAVITATIONAL
the equations of motion
RADIATION
95
DAMPING
(11) which are most conveniently
written as
= 0,
(Qx");,
or (3) The total rate of work done by t)he system is given by u=-
/
(Q%‘“)
,y dS, = - /
(Q%“‘),,
dV = -;
j
Qooo
dV,
(4)
since the surface integrals arising in the third member from the use of Gauss’ theorem can be taken to vanish as a consequence of the assumption that the system is localized. Let us assume that the gravitational field is weak. This means that the quantities - ,,‘* yw = (5) Qpy
along with their derivatives can be considered to be sufficiently small so that their squares and products can be neglected. It follows that the yPy can be taken to satisfy the equations MS Y ,Y = 0, (la! and fy”*
- ?P” = - lfjnT”,
(2%)
where, for convenience we let T”” = Q%'". However, in view of our assumption that the gravitational forces are weak compared to the other forces present, one can replace any Q@occurring in T’” by qa8 to a sufficiently good approximation, i.e., one can take T”’ to be independent of Q@ and therefore known. The rate of work U, on the basis of (3) and the weak field approximation, can now be written:
- 3~“) ,o Too - yoo,k Tok (7)
1 :!
+ 2yak,l + f 6kL(yoo - ynn),O Tkl
dV.
To evaluate (7) we solve (2a) for y’“, taking the retarded potential
solution
96
PERES
-/“(r,t)
= 4
AND
ROSEN
T”“(r’, t - R) dV’/R,
s
(8)
where R = ) r - r’j,
(9)
and then substitute into (7). We now expand T”‘(r’, t - R) in powers of R, T““(r’, t - R) = T’“’ - ‘j’“‘R
+fivR2+‘pvR3+
.. . .
(10)
where T’“’ = T”( r’, t). It will be assumed that such an expansion is valid. This imposes restrictions on the time derivatives of T”. For example, in the case of material motion, this implies that the velocities are small. Substituting (10) into (8) and integrating term by term, we get - ,‘@’ + i;‘@‘&f2!
Y” = 4 $ [T’“/R
-
“1;I”R2/3!
+ . . .] dJ7’.
This is to be substituted into Eq. (7). We now write down a number of relations which simplify the calculation. the definition of T”“, we have T@“,” = 0, to the required accuracy, or
(11)
From
+@’ = - TMktk ,
(12)
/- !i”’ dV = - 1 Tpk d& = 0,
(13)
whence
since the system is spatially
bounded. In general one has
/ AB,I, dV = - $ A,kBdV,
(14)
where A and B are functions such that AB falls off sufficiently rapidly at large distances. It follows that s
TpOxk ,jV = -
J
T”$sk
,-jV =
s
T’” dV,
j+‘%“dV=/+dV=O,
s
+‘zkd
In the calculation
dV =
s
Tmn,m,zkzl dV = 2
(15) (16)
s
Tki dV.
(17)
of the rate of work it will be understood that the average is
GRAVITATIONAL
RADIATION
97
DAMPIKG
to be taken over a long time interval. From the assumption that the system remains in a finite volume it follows that in the approximation needed for evaluating the right-hand side of (7)) one can take effectively that
where F is any function of the field sources, in the sense that the average value of the derivative over a long time interval vanishes to this approximation. In the same sense we can write
where A and B are two functions of the sources. By the use of (14) and (19) one gets from (7) yoo + ykk),oTm
- yok,oTok
11
(20)
Tk2 dV. ,O
Making use of (18)) one finds as might have been expected, that only the odd terms in (11) contribute to U, that is, the terms occurring in t’he “radiation field” (12) :
+;i;““&l3! +‘T““‘R”/5! +...]dv’.(21) $;;‘, =-4s@“
With respect to velocities (or time derivatives), considered as small quantities of the first order, Too is of order 0, Tok of order 1, and Tkt (aside from possibly a static part) of order 2. One has then from (11) and ( 13 j, .. . 4 (22) Too = -j-j .s T’OOR’ dV’, = -4 Ok
=
-;
=
--
J j
plkl dv’, ;;;"','
(33) dv',
(24)
?;
00 Y
.....
4
6
5!
T’O”R4 dV’,
(“5)
s
... 4 Ttk1R2 dV’. (26) 3! s The lowest order terms in U are found to vanish. For example, the terms in kl
Y 5
zr
--
98
PERES
(20) corresponding
AND
to dipole radiation
yoo+ T’~),,,T~
ROSEN
are given by dV = -1 4
s
,yoo + ykk,,cx, dV.
(27)
However, one can easily see from (17), (22), and (23) that 1;” and :” are functions only of time. They can be taken outside the last integral in (27) which therefore vanishes, because of (13). Thus there is no gravitational dipole radiation. The first generally nonvanishing terms are those corresponding to quadrupole radiation : f
+ ~kk),c,Too -
lok,oTok
1I Tkl ,O
(28)
dV.
By the help of the previous relations this can be put into the form
(29) If we introduce Dkl = 3 1 Tooxkxz dV - 8kl / Tmxnxn dV,
(30)
u = & &,ij;k,,
(31)
we get 6
in agreement with the results of the linear theory for the rate of radiation from a physical system (1). The present method of calculation, similar to that of Lorentz in his treatment of the electron (13)) has the advantage that it does not make use of the behavior of the field at infinity where difficulties may arise with the boundary conditions when one takes nonlinear terms into account (14). The agreement of the results obtained by different methods indicates an inner consistency in the theory of gravitational radiation and is a strong argument in support of the standpoint that gravitational waves have a real existence in the framework of the general theory of relativity. RECEIVED:
November
1. L.LANDAUAND
Cambridge,
10,1959.
E. LIFSHITZ, 1951.
REFERENCES “The Classical Theory of Fields,”
p. 331. Addison-Wesley,
GRAVITATIONAL
2. S. 4. 5. 6. 7. 8. 9. 10. If. f2. 13. 14.
N. L. A. P. A. A. I,. A. W. F. A. H. A.
RSDIATION
DAMPIKG
99
Hu, Proc. Roy. Irish Acad. AM, 87 (1947). INFELD AND A. E. SCHEIDEGGER, Can. J. Math. 3, 195 (1951). E. SCHEIDEGGER, Phys. Rev. 82, 883 (1951); Revs. Modern Phys. 26, 451 (1953). HAVAS, Phys. Rev. 108, 1351 (1957). TRAUTMAN, Bu.11. Acad. Polonaise Sci. 6, 627 (1958). PERES, Nuovo cimenlo 11, 644 (1959). INFELD, Ann. Phys. 6, 341 (1959). PERES, Nuovo cinrento 13,670 (1959). B. BONNOR, Nature 181,1196 (1958); Phil. Trans. Roy. Sot. London A%%,233 (1959). HENNEQUIN, thesis, University of Paris, 1956. PERES, Nuovo cimento 11, 617 (1959). LORIXNTZ, “The Theory of Electrons,” p. 254. Dover, New Pork, 1952. PJCRM AND N. ROSEN, Phys. Rev. 116, 1085 (1959).
OF
PHYSICS:
10,
94-99
(1960)
Gravitational
Radiation
Damping
Nongravitational
Motion*
ASHER PERES AND NATHAN Department
of Physics,
Israel
of
Institute
ROSEN
of Technology,
Haifa,
Israel
A material system acted upon by internal forces much larger than the gravitational ones is considered, and the rate of work of this system against its own gravitational field is computed. The result is equal to the rate of radiated energy obtained from the linearized theory.
The existence of gravitational radiation has long been a controversial subject. It seemedwell established according to the wave zone approximation of the linearized theory (I ), but the investigation of the equations of motion (Z-8) led to very dif7erent results. Only recently, an agreement between both methods was reached in the case of freely gravitating pole particles (9) and in a special case of constrained motion (IO). Our purpose in the present paper is to show that if a system is acted on by internal forces much larger than those due to gravitation, its rate of work against its own gravitational field (averaged over a cycle of the motion or over a long time interval) is equal to the rate of loss of energy by gravitational radiation, as calculated by means of the linear approximation. The proof will be rather general, i.e., independent of the nature of the motion (rotational, vibrational, or other), but with the condition that the system remain localized within a finite volume. We take as field variables the contravariant densities Q’“, and choose quasiGalilean coordinates (Q” - q’“) subject to the harmonic conditionl: PY Q ,Y - 0. (1) The Einstein field equations then have the form, in natural units, v2Q”y _ ‘;1’Y = -16~gX:“”
+ O”‘,
(2)
where 5“’ is the stress-energy tensor density, Q = (-det. Q’“)~‘~, and 0’” is quadratic in (9”’ - r]“). The compatibility of Eqs. (1) and (2) follows from * Partly supported by the U. S. Air Force, through search and Development Command. 1 Greek indices run from 0 to 3, Latin indices-from 94
the European 1 to 3.
Office of the Air Re-
GRAVITATIONAL
the equations of motion
RADIATION
95
DAMPING
(11) which are most conveniently
written as
= 0,
(Qx");,
or (3) The total rate of work done by t)he system is given by u=-
/
(Q%‘“)
,y dS, = - /
(Q%“‘),,
dV = -;
j
Qooo
dV,
(4)
since the surface integrals arising in the third member from the use of Gauss’ theorem can be taken to vanish as a consequence of the assumption that the system is localized. Let us assume that the gravitational field is weak. This means that the quantities - ,,‘* yw = (5) Qpy
along with their derivatives can be considered to be sufficiently small so that their squares and products can be neglected. It follows that the yPy can be taken to satisfy the equations MS Y ,Y = 0, (la! and fy”*
- ?P” = - lfjnT”,
(2%)
where, for convenience we let T”” = Q%'". However, in view of our assumption that the gravitational forces are weak compared to the other forces present, one can replace any Q@occurring in T’” by qa8 to a sufficiently good approximation, i.e., one can take T”’ to be independent of Q@ and therefore known. The rate of work U, on the basis of (3) and the weak field approximation, can now be written:
- 3~“) ,o Too - yoo,k Tok (7)
1 :!
+ 2yak,l + f 6kL(yoo - ynn),O Tkl
dV.
To evaluate (7) we solve (2a) for y’“, taking the retarded potential
solution
96
PERES
-/“(r,t)
= 4
AND
ROSEN
T”“(r’, t - R) dV’/R,
s
(8)
where R = ) r - r’j,
(9)
and then substitute into (7). We now expand T”‘(r’, t - R) in powers of R, T““(r’, t - R) = T’“’ - ‘j’“‘R
+fivR2+‘pvR3+
.. . .
(10)
where T’“’ = T”( r’, t). It will be assumed that such an expansion is valid. This imposes restrictions on the time derivatives of T”. For example, in the case of material motion, this implies that the velocities are small. Substituting (10) into (8) and integrating term by term, we get - ,‘@’ + i;‘@‘&f2!
Y” = 4 $ [T’“/R
-
“1;I”R2/3!
+ . . .] dJ7’.
This is to be substituted into Eq. (7). We now write down a number of relations which simplify the calculation. the definition of T”“, we have T@“,” = 0, to the required accuracy, or
(11)
From
+@’ = - TMktk ,
(12)
/- !i”’ dV = - 1 Tpk d& = 0,
(13)
whence
since the system is spatially
bounded. In general one has
/ AB,I, dV = - $ A,kBdV,
(14)
where A and B are functions such that AB falls off sufficiently rapidly at large distances. It follows that s
TpOxk ,jV = -
J
T”$sk
,-jV =
s
T’” dV,
j+‘%“dV=/+dV=O,
s
+‘zkd
In the calculation
dV =
s
Tmn,m,zkzl dV = 2
(15) (16)
s
Tki dV.
(17)
of the rate of work it will be understood that the average is
GRAVITATIONAL
RADIATION
97
DAMPIKG
to be taken over a long time interval. From the assumption that the system remains in a finite volume it follows that in the approximation needed for evaluating the right-hand side of (7)) one can take effectively that
where F is any function of the field sources, in the sense that the average value of the derivative over a long time interval vanishes to this approximation. In the same sense we can write
where A and B are two functions of the sources. By the use of (14) and (19) one gets from (7) yoo + ykk),oTm
- yok,oTok
11
(20)
Tk2 dV. ,O
Making use of (18)) one finds as might have been expected, that only the odd terms in (11) contribute to U, that is, the terms occurring in t’he “radiation field” (12) :
+;i;““&l3! +‘T““‘R”/5! +...]dv’.(21) $;;‘, =-4s@“
With respect to velocities (or time derivatives), considered as small quantities of the first order, Too is of order 0, Tok of order 1, and Tkt (aside from possibly a static part) of order 2. One has then from (11) and ( 13 j, .. . 4 (22) Too = -j-j .s T’OOR’ dV’, = -4 Ok
=
-;
=
--
J j
plkl dv’, ;;;"','
(33) dv',
(24)
?;
00 Y
.....
4
6
5!
T’O”R4 dV’,
(“5)
s
... 4 Ttk1R2 dV’. (26) 3! s The lowest order terms in U are found to vanish. For example, the terms in kl
Y 5
zr
--
98
PERES
(20) corresponding
AND
to dipole radiation
yoo+ T’~),,,T~
ROSEN
are given by dV = -1 4
s
,yoo + ykk,,cx, dV.
(27)
However, one can easily see from (17), (22), and (23) that 1;” and :” are functions only of time. They can be taken outside the last integral in (27) which therefore vanishes, because of (13). Thus there is no gravitational dipole radiation. The first generally nonvanishing terms are those corresponding to quadrupole radiation : f
+ ~kk),c,Too -
lok,oTok
1I Tkl ,O
(28)
dV.
By the help of the previous relations this can be put into the form
(29) If we introduce Dkl = 3 1 Tooxkxz dV - 8kl / Tmxnxn dV,
(30)
u = & &,ij;k,,
(31)
we get 6
in agreement with the results of the linear theory for the rate of radiation from a physical system (1). The present method of calculation, similar to that of Lorentz in his treatment of the electron (13)) has the advantage that it does not make use of the behavior of the field at infinity where difficulties may arise with the boundary conditions when one takes nonlinear terms into account (14). The agreement of the results obtained by different methods indicates an inner consistency in the theory of gravitational radiation and is a strong argument in support of the standpoint that gravitational waves have a real existence in the framework of the general theory of relativity. RECEIVED:
November
1. L.LANDAUAND
Cambridge,
10,1959.
E. LIFSHITZ, 1951.
REFERENCES “The Classical Theory of Fields,”
p. 331. Addison-Wesley,
GRAVITATIONAL
2. S. 4. 5. 6. 7. 8. 9. 10. If. f2. 13. 14.
N. L. A. P. A. A. I,. A. W. F. A. H. A.
RSDIATION
DAMPIKG
99
Hu, Proc. Roy. Irish Acad. AM, 87 (1947). INFELD AND A. E. SCHEIDEGGER, Can. J. Math. 3, 195 (1951). E. SCHEIDEGGER, Phys. Rev. 82, 883 (1951); Revs. Modern Phys. 26, 451 (1953). HAVAS, Phys. Rev. 108, 1351 (1957). TRAUTMAN, Bu.11. Acad. Polonaise Sci. 6, 627 (1958). PERES, Nuovo cimenlo 11, 644 (1959). INFELD, Ann. Phys. 6, 341 (1959). PERES, Nuovo cinrento 13,670 (1959). B. BONNOR, Nature 181,1196 (1958); Phil. Trans. Roy. Sot. London A%%,233 (1959). HENNEQUIN, thesis, University of Paris, 1956. PERES, Nuovo cimento 11, 617 (1959). LORIXNTZ, “The Theory of Electrons,” p. 254. Dover, New Pork, 1952. PJCRM AND N. ROSEN, Phys. Rev. 116, 1085 (1959).